In this paper, we continue to develop and study the conservative level set method for incompressible two phase flow with surface tension introduced in [J. Comput. Phys. 210 (2005) 225-246]. We formulate a modification of the reinitialization and present a theoretical study of what kind of conservation we can expect of the method. A finite element discretization is presented as well as an adaptive mesh control procedure. Numerical experiments relevant for problems in petroleum engineering and material science are presented. For these problems the surface tension is strong and conservation of mass is important. Problems in both two and three dimensions with uniform as well as non-uniform grids are studied. From these calculations convergence and conservation is studied. Good conservation and convergence are observed.

We present a finite element method for the Stokes equations involving two immiscible incompressible fluids with different viscosities and with surface tension. The interface separating the two fluids does not need to align with the mesh. We propose a Nitsche formulation which allows for discontinuities along the interface with optimal a priori error estimates. A stabilization procedure is included which ensures that the method produces a well conditioned stiffness matrix independent of the location of the interface.Keywords cut finite element method, CutFEM · Nitsche's method · two-phase flow · discontinuous viscosity · surface tension · sharp interface method 1 Introduction

We develop a cut finite element method for a second order elliptic coupled bulksurface model problem. We prove a priori estimates for the energy and L 2 norms of the error. Using stabilization terms we show that the resulting algebraic system of equations has a similar condition number as a standard fitted finite element method. Finally, we present a numerical example illustrating the accuracy and the robustness of our approach.

In this contribution we present a new computational method for coupled bulk-surface problems on time-dependent domains. The method is based on a space-time formulation using discontinuous piecewise linear elements in time and continuous piecewise linear elements in space on a fixed background mesh. The domain is represented using a piecewise linear level set function on the background mesh and a cut finite element method is used to discretize the bulk and surface problems. In the cut finite element method the bilinear forms associated with the weak formulation of the problem are directly evaluated on the bulk domain and the surface defined by the level set, essentially using the restrictions of the piecewise linear functions to the computational domain. In addition a stabilization term is added to stabilize convection as well as the resulting algebraic system that is solved in each time step. We show in numerical examples that the resulting method is accurate and stable and results in well conditioned algebraic systems independent of the position of the interface relative to the background mesh.